## Free modal algebras: a coalgebraic perspective

Citations: | 5 - 1 self |

### BibTeX

@MISC{Bezhanishvili_freemodal,

author = {N. Bezhanishvili and A. Kurz},

title = {Free modal algebras: a coalgebraic perspective},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1

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Citation Context ...nal logic is to algebras. The precise relationship between the logics and the coalgebras can be formulated using Stone duality [9]. From this perspective, algebras are the logical forms of coalgebras =-=[1]-=-; and the algebras that appear in this way give rise to modal logics. In this paper we take the opposite view and ask how coalgebraic and categorical methods can elucidate traditional topics in modal ... |

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Citation Context ...∈ Xk+1 can be understood as a tree with the root labelled by an element l ∈ X0 and the children being the elements of S ∈ P(Xk). These trees have a rich history and have been studied, for example, by =-=[3, 2, 6, 18, 5, 31]-=-. Corollary 5.4. The modal space (Xω, Rω), where Xω is the limit in Stone of the family {Xk}k∈ω with the maps πk+1 : Xk+1 → Xk, and Rω is defined by (xi)i∈ωRω(yi)i∈ω if xk+1Rkyk for each k ∈ ω is (iso... |

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Citation Context ...ebras for the functor V and, therefore, the free modal algebras can be obtained by a standard construction in category theory, the initial algebra sequence. Indeed, under fairly general circumstances =-=[4]-=-, for a functor L on a category C, the L-algebra Lω free over C ∈ C is the colimit of the sequence (Ln)n<ω L0 e0 �� L1 e1 �� L2 . . . Lω (1) where L0 = 0 is the initial object of C and Ln+1 = (C + L)(... |

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Citation Context ...ar, it has been recognised that modal logic is to coalgebras what equational logic is to algebras. The precise relationship between the logics and the coalgebras can be formulated using Stone duality =-=[9]-=-. From this perspective, algebras are the logical forms of coalgebras [1]; and the algebras that appear in this way give rise to modal logics. In this paper we take the opposite view and ask how coalg... |

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Citation Context ...s of modal logics and for deriving completeness results for these logics. Moss [25] revisited Fine’s formulas to give a filtration type finite-model property proofs for various modal logics. Abramsky =-=[2]-=- constructed the canonical model of closed formulas of the basic modal logic as the final coalgebra for the Vietoris functor and Ghilardi [18, 17] gave a similar description of canonical models of mod... |

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Citation Context ... in the coalgebraic representation of modal spaces as coalgebras for the Vietoris functor [22] and in the coalgebraic representation of modal Priestley spaces as coalgebras for the convex set functor =-=[20, 27]-=-. This allows us to represent the canonical models of modal and positive modal logic as a limit of finite sets and posets, respectively. We also observe that the underlying Stone space of the canonica... |

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Citation Context ...emporal algebra. 1 Introduction Modal logics play an important role in many areas of computer science. In recent years, the connection of modal logic and coalgebra received a lot of attention, see eg =-=[30]-=-. In particular, it has been recognised that modal logic is to coalgebras what equational logic is to algebras. The precise relationship between the logics and the coalgebras can be formulated using S... |

10 |
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Citation Context ... x ∈ X the set Q[x] is a closed upset of Y (resp. a closed downset of Y) and for every clopen upset U of Y the set [Q]U is a clopen upset of X (resp. 〈Q〉U is a clopen downset of X). Theorem 2.11. (eg =-=[11]-=-) There is a one-to-one correspondence between join preserving (resp. meet preserving) maps between distributive lattices and clopen increasing (resp. clopen decreasing) relations on their dual Priest... |

10 |
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Citation Context ...e-model property proofs for various modal logics. Abramsky [2] constructed the canonical model of closed formulas of the basic modal logic as the final coalgebra for the Vietoris functor and Ghilardi =-=[18, 17]-=- gave a similar description of canonical models of modal and intuitionistic logics to derive a normal form representation for these logics. For positive modal logic similar techniques were developed b... |

10 |
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Citation Context ...∈ Xk+1 can be understood as a tree with the root labelled by an element l ∈ X0 and the children being the elements of S ∈ P(Xk). These trees have a rich history and have been studied, for example, by =-=[3, 2, 6, 18, 5, 31]-=-. Corollary 5.4. The modal space (Xω, Rω), where Xω is the limit in Stone of the family {Xk}k∈ω with the maps πk+1 : Xk+1 → Xk, and Rω is defined by (xi)i∈ωRω(yi)i∈ω if xk+1Rkyk for each k ∈ ω is (iso... |

8 |
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Citation Context ...nonical model of a given modal logic can be the key for understanding the properties of this logic. The general idea that we will discuss in this paper has appeared before in different contexts. Fine =-=[16]-=- used his canonical formulas for describing canonical models of modal logics and for deriving completeness results for these logics. Moss [25] revisited Fine’s formulas to give a filtration type finit... |

7 | STS: A structural theory of sets, L
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Citation Context ...∈ Xk+1 can be understood as a tree with the root labelled by an element l ∈ X0 and the children being the elements of S ∈ P(Xk). These trees have a rich history and have been studied, for example, by =-=[3, 2, 6, 18, 5, 31]-=-. Corollary 5.4. The modal space (Xω, Rω), where Xω is the limit in Stone of the family {Xk}k∈ω with the maps πk+1 : Xk+1 → Xk, and Rω is defined by (xi)i∈ωRω(yi)i∈ω if xk+1Rkyk for each k ∈ ω is (iso... |

7 |
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Citation Context ...e-model property proofs for various modal logics. Abramsky [2] constructed the canonical model of closed formulas of the basic modal logic as the final coalgebra for the Vietoris functor and Ghilardi =-=[18, 17]-=- gave a similar description of canonical models of modal and intuitionistic logics to derive a normal form representation for these logics. For positive modal logic similar techniques were developed b... |

6 |
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(Show Context)
Citation Context ...his paper has appeared before in different contexts. Fine [16] used his canonical formulas for describing canonical models of modal logics and for deriving completeness results for these logics. Moss =-=[25]-=- revisited Fine’s formulas to give a filtration type finite-model property proofs for various modal logics. Abramsky [2] constructed the canonical model of closed formulas of the basic modal logic as ... |

6 |
A coalgebraic view on positive modal logic
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(Show Context)
Citation Context ... in the coalgebraic representation of modal spaces as coalgebras for the Vietoris functor [22] and in the coalgebraic representation of modal Priestley spaces as coalgebras for the convex set functor =-=[20, 27]-=-. This allows us to represent the canonical models of modal and positive modal logic as a limit of finite sets and posets, respectively. We also observe that the underlying Stone space of the canonica... |

4 | A complete coalgebraic logic - Kupke, Kurz, et al. - 2008 |

3 | An elementary construction for a non-elementary procedure. Studia Logica
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Citation Context ...ralise to all logics of rank 1 (as long as the axioms are effectively given). This should be related to recent work of Schröder and Pattinson [29] on the complexity of rank 1 logics. Marx and Mikulás =-=[24]-=- also obtain complexity bounds for bi-modal logics by looking into algebras of terms of degree ≤ k. Obtaining normal forms for logics that are not axiomatised by formulas of rank 1 is another interest... |

2 | Distributive lattice-structured ontologies - Bruun, Coumans, et al. - 2009 |

2 |
A Remark of Spaces 2 X for Zero-dimensional X
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Citation Context ...e basic modal logic is homeomorphic to the so-called Pelczynski space. This space appears to be one of the nine fixed points of the Vietoris functor on compact Hausdorff spaces with a countable basis =-=[28, 26]-=-. As we will see below, this method directly applies to modal and positive modal logics that are axiomatised by the formulas of rank 1. We also indicate how to adjust our techniques to modal logics th... |

1 |
The free p-algebra generated by a distributive lattice
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Citation Context ...iption of canonical models of modal and intuitionistic logics to derive a normal form representation for these logics. For positive modal logic similar techniques were developed by Davey and Goldberg =-=[13]-=-. ⋆ Supported by the EPSRC grant EP/C014014/1 and by the Georgian National Science Foundation grant GNSF/ST06/3-003. ⋆⋆ Partially supported by EPSRC EP/C014014/1.sThe aim of this paper is to unify all... |

1 |
Strongly complete logics for coalgebras. Submitted, electronically available
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Citation Context ...p → ♦p and ♦♦p → ♦p are not. The precise 4 For us, a variety is given by operations of finite arity and equations.srelationship between L-algebras and algebras for an extended signature is studied in =-=[23]-=-. Roughly speaking, there is a one-to-one correspondence between functors L : V → V and extensions of V by operations Σ ′ and equations of rank 1 E ′ ; under this correspondence, Alg(L) is isomorphic ... |

1 |
The topological types of hyperspaces of 0-dimensional compacta. Topology and its Applications
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- 2005
(Show Context)
Citation Context ...e basic modal logic is homeomorphic to the so-called Pelczynski space. This space appears to be one of the nine fixed points of the Vietoris functor on compact Hausdorff spaces with a countable basis =-=[28, 26]-=-. As we will see below, this method directly applies to modal and positive modal logics that are axiomatised by the formulas of rank 1. We also indicate how to adjust our techniques to modal logics th... |

1 | PSPACE bounds for rank 1 modal logics
- Schröder, Pattinson
- 2006
(Show Context)
Citation Context ... to Xk+1. The procedure to obtain normal forms should generalise to all logics of rank 1 (as long as the axioms are effectively given). This should be related to recent work of Schröder and Pattinson =-=[29]-=- on the complexity of rank 1 logics. Marx and Mikulás [24] also obtain complexity bounds for bi-modal logics by looking into algebras of terms of degree ≤ k. Obtaining normal forms for logics that are... |